A circle in the coordinate plane passes through points (-3, -2) and (1

VERITAS PREP OFFICIAL SOLUTION:

This problem requires you to leverage the definition that the longest chord in any circle is the diameter. Since the line connecting the two given points on the circle, (-3, -2) and (1, 4), is a chord, the smallest possible circle would occur if that chord were the diameter (the longest possible line).

Given that, taking the distance between (-3, -2) and (1, 4) will provide you with the diameter of that smallest possible circle. Since the horizontal difference (between x-coordinates) is 4 and the vertical difference (between y-coordinates) is 6, you can calculate the distance using Pythagorean Theorem:

4^2+6^2=c^2, so c^2=52 and \(c=\sqrt{52}=2\sqrt{13}\).

Remember that this distance is the diameter, so to find the area you'll want to cut it in half to find the radius. Therefore the radius is \(\sqrt{13}\) and \(\pi r^2=\pi (\sqrt{13})^2=13\pi\).

A circle in the coordinate plane passes through points (-3, -2) and (1, 4). What is the smallest possible area of that circle?

The smallest possible area for the circle will be one that has the distance between the 2 aforementioned points as the diameter.

The distance between the two points is sqrt(52).
radius = sqrt(52)/2
Area= pi*(sqrt(52)/2)^2

Answer:
A. 13π

Let me try. When you draw series of circles through the points (-3,-2) and *(1,4), you will see that the line connecting these 2 points can be either a chord or the diameter (refer to the attached figure)

If the line is a chord, then the radius of this circle will be > the distance between the 2 points (=\(2\sqrt {13}\))

But, if the line is the diameter of the circle, then the distance = diameter of the circle. As area of a circle is dependent on the diameter of the circle, the smallest area of the circle will be when the line joining the 2 points above is the diameter of the circle.

Thus, the circle with minimum area will have a diameter = \(2\sqrt {13}\) and thus Area = \(\frac{\pi *(2\sqrt {13})^2}{4}\) = \(13*\pi\)

Answer is 13pi.

The minimum area would be if these points are the endpoints of a diameter. using the distance formula or constructing a triangle and calculating the value of hypotenuse, we get the value of distance of the diameter and accordingly calculate the value of area.

I selected A. but sorry Bunuel, I can not understand your point here. is there any drawing. I can not imagine. Sorry again.

Great analysis. thank you pretty much Engr2012.

Distance = Diameter = \(\sqrt{(4+2)^2 + (1+3)^2}=>\sqrt{52}=2\sqrt{13}\)

Thus, Radius \(= \sqrt{13}\)

Hence area of the circle \(= π*\sqrt{13}*\sqrt{13} = 13π\), Answer must be (A)

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